The Fundamental Lemma for central simple algebras

Let $D$ be a central simple algebra and let $G$ and $G'$ be the groups in the Guo–Jacquet comparison, with test functions $f_D=1_{O_D^\times}$ and $f'_D$ as defined from the standard lattice chain. For regular semi-simple $\gamma\in G'_{\mathrm{rs}}$, matching means that $\gamma$ and $g\in G_{\mathrm{rs}}$ have the same relative invariants. Fundamental Lemma for central simple algebras. The function $f'_D$ is a transfer of $f_D$:

$$\operatorname{Orb}(\gamma, f'_D)=\begin{cases}\operatorname{Orb}(g,f_D)&\text{if there exists a matching }g\in G_{\mathrm{rs}},\\0&\text{otherwise.}\end{cases}$$

This extends the Guo–Jacquet Fundamental Lemma from the split case $D=M_{2n}(F)$ to central simple algebras. The split case for the full Hecke algebra was proved by Guo, while the general assertion is the conjectural part.